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Baryshnikov, Yuliy Hirani, Anil Schenck, Hal Yong, Alexander Kosar, Nicholas J 2018-09-04T20:33:59Z 2018-09-04T20:33:59Z 2020-09-05T09:15:16Z 2018-04-10 2018-05 We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the $n^{th}$ no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of $\N^m$. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of $\R^d$. Next, we consider the relation between no $k$-equal spaces of $\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the $n^{th}$ no $k$-equal space of $\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in $\R^d$. Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-05-01 The student, Nicholas Kosar, accepted the attached license on 2018-04-07 at 12:04. The student, Nicholas Kosar, submitted this Dissertation for approval on 2018-04-07 at 12:05. This Dissertation was approved for publication on 2018-04-10 at 08:04. DSpace SAF Submission Ingestion Package generated from Vireo submission #12146 on 2018-08-31 at 17:18:16 Made available in DSpace on 2018-09-04T20:33:59Z (GMT). No. of bitstreams: 2 KOSAR-DISSERTATION-2018.pdf: 535418 bytes, checksum: de4bddc35e057da1ef4ff274491c039d (MD5) LICENSE.txt: 4211 bytes, checksum: 2c07369115c366140909668669b07ab9 (MD5) Previous issue date: 2018-04-10 Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:34:13Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:37:00Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system Embargo set by: Seth Robbins for item 107225 Lift date: 2020-09-04T20:42:08Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system U of I Only Restriction Lifted for Item 107225 on 2020-09-05T09:15:16Z. application/pdf http://hdl.handle.net/2142/101142 en Copyright 2018 Nicholas Kosar Polychromatic configuration spaces k-spanning trees discriminantal arrangements Generalizations of no k-equal spaces text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.